Distribution of the generalised inverse of a random matrix and its applications

Dı́az-Garcı́a, José A.; Jáimez, Ramón Gutiérrez

doi:10.1016/j.jspi.2004.06.032

Cited by 28 publications

(25 citation statements)

References 5 publications

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“…We note that the similar results as our Lemmas 2 and 3 have been independently proposed in [2,3]. The main difference is in that the derivations of [2,3] are based on exterior products, whereas ours based on the singular value decomposition. In addition, since Theorem 1 of [3] requires that the matrices involved are positive semidefinite, this theorem is a special case of Lemma 3.…”

Section: Lemmasupporting

confidence: 75%

“…If X is symmetric and nonsingular, i.e., r=n, then J (Y → X) = |X| −(n+1) = |Y| n+1 [5, p.14]. We note that the similar results as our Lemmas 2 and 3 have been independently proposed in [2,3]. The main difference is in that the derivations of [2,3] are based on exterior products, whereas ours based on the singular value decomposition.…”

Section: Lemmamentioning

confidence: 52%

“…Interested readers are referred to [7, p. 34-36] for more details. Throughout this paper, we assume that X = 0 due to Proposition 1 (3). In this section, we consider a simple case.…”

Section: Pseudo-inverse Multivariate Distributionsmentioning

confidence: 99%

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Pseudo-inverse multivariate/matrix-variate distributions

Zhang

2007

Journal of Multivariate Analysis

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The Moore-Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore-Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore-Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudoinverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.

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Section: Lemmasupporting

confidence: 75%

Section: Lemmamentioning

confidence: 52%

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Pseudo-inverse multivariate/matrix-variate distributions

Zhang

2007

Journal of Multivariate Analysis

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“…However, few are the works published in the singular case, in fact, only after the papers of Uhlig [25] and Díaz-García et al [11], the articles on Jacobian computations of singular matrices began to be multiplied thanks to use of the forgotten technique of James [13]; see Díaz-García and Gutiérrez [8], Díaz-García and Gutiérrez-Jáimez [10], Díaz-García and González-Farías [6,7]. Another method applied to the singular case is explained in Srivastava [24].…”

Section: Introductionmentioning

confidence: 99%

A note about measures and Jacobians of singular random matrices

Dı́az-Garcı́a

2007

Journal of Multivariate Analysis

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This paper explains the differences between the densities and the Jacobians of the transforms of the same singular random matrices treated by several authors. Some comments on the results proposed by Srivastava [Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003) 1537-1560] are presented. Definitions about a measure with respect to which a singular random matrix possesses a density are proposed. Finally two Jacobians of certain transforms under any of those measures are found.

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“…Also, some related singular distributions have been proposed: the Wishart distribution, the matrix variate T, the matrix variate beta type I and II, among many others, see Díaz-García and González-Farías (2005a,b), , Díaz-García and Gutiérrez-Jáimez (2006), Díaz-García, Gutiérrez-Jáimez, and Mardia (1997), Ip, Wong, and Liu (2007) and Uhlig (1994).…”

Section: Introductionmentioning

confidence: 99%

Singular extended skew-elliptical distributions

Dı́az-Garcı́a

González–Farías

2008

Journal of the Korean Statistical Society

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Singular vector and matrix extended skew-elliptical distributions are studied in this work. Based on the vectorial case, two alternatives for singular matrix variate extended skew-elliptical distribution are also proposed. In addition, the distributions of a general linear transformation for extended skew-elliptical vectors and matrices are derived along with the corresponding density functions. These results are applied in the distribution of the residuals for a general linear model with extended skew-elliptical errors.

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Distribution of the generalised inverse of a random matrix and its applications

Cited by 28 publications

References 5 publications

Pseudo-inverse multivariate/matrix-variate distributions

Pseudo-inverse multivariate/matrix-variate distributions

A note about measures and Jacobians of singular random matrices

Singular extended skew-elliptical distributions

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